Those retirement fund problems don't go away if you privatize it, in fact they get a lot worse since there is no longer a direct revenue stream to pay any of it, and that government obligation either has to come out of general government revenue (I.e., taxes) or whatever private provider picks up the service has to charge enough on top of its own costs to put those funds into it.
It could easily pay for itself, the problem is whenever the USPS starts to make noise about charging what it actually costs, every pennypinching old gaffer, little old widow, and mass mailer writes angry letters to their Congresscritter, and Congress proceeds to crap all over any plan to relate revenue to the actual costs. The cost of a first class letter in the US is a loss leader price compared to what it costs for the equivalent service from the Royal Mail or Bundespost.
It's a Constitutionally mandated service, and although privatizing it is a perennial pet rock of the GOP, not that many people outside the shipping industry lobbyists really sees a real need to do that. It's not even part of the original Constitutional concept that it had to pay for itself, the Constitution does not address that. If the service characteristics remain the same, i.e. routine daily residential pickup and delivery, it isn't going to be any cheaper from the user perspective. The talking heads on Fox Business and the other financial channels get moist talking about it, but they've all got a NYC/Beltway-centric life and POV, where UPS, FEDEX, DHL, bicycle couriers etc. are available any time of day. It's a whole different world away from the metroplexes, where there are one or two delivery services with facilities within an hour's drive, and there might be a couple of pickup locations in the entire county, or if they're lucky a retail outlets that charge on top of the shipper's rates for service.
